How do we find a *new* tone which is not a duplication? We know such a tone cannot be
an even number. Since we have started *counting* with our first two tones, 1 and 2, we
simply keep counting, and arrive at the number 3, which is the first *odd prime*. If the
string is touched and bowed at a point *one third* of the way along the neck of the
viola, a *new* pitch is produced.

This is called the *third harmonic* in the harmonic series. The three arcs below
show that the string is broken into thirds.

Although this new pitch is clearly related to the pitch produced by the entire string,
it is also clearly not a duplication of it. The vibrating segment is *one third* as
long as the whole string, so we can call this segment 1/3. This segment also vibrates 3
times as fast as the first tone. The figure above is shown in blue to signify that the number
3 represents *the first new tone*. The value of this tone, 3, is obviously larger than 2.
To justify this tone, we use the relationship of duplication, dividing the tone by 2, which
gives us 3/2, a congruence of 3 which sounds *lower* in pitch.

Having justified this new tone, we can see how it fits within the boundary tones. The figure below signifies that the new tone was derived from the source tone. The number line below the figure reflects how we hear the new tone in terms of its distance from the boundary tones.

3/2 is equal to 1 and 1/2, or 1.5 stated as a decimal fraction, shown below on the common linear number line.

As the first *new* tone, the tone 3/2 has a prominant function within the structure of the
Western system. The significance of this tone is reflected in the name given to it by music
theorists; it is called the *dominant*.

The figure below summarizes the names of the tones we have derived so far.

The third harmonic expanded the system with a new tone. How are more tones added? First, we simply
try counting. We reach the number 4, bringing us to the *fourth harmonic* in the
*harmonic series*, which may be produced by touching and bowing the string at *one fourth*
of its entire length.

The arcs show the string broken into four sections. Because 4 is 2 x 2, this tone is a duplication of the second harmonic, which in turn was a duplication of the first harmonic, or fundamental. So, unfortunately this tone does not expand the system.

Symmetry is an appealing property of many natural forms. So far, we have derived an asymmetrical system, as shown below.

A tone must be added to make the system symmetrical. How is this done? Consider that the relationship between the first new tone 3 and the first tone 1 is 3/1, which becomes 3/2 when placed within the boundaries of 1 and 2. If we use this relationship in reverse, we create symmetry around the source tone. This new tone is not produced by a segment 1/3 as long as the entire string, but rather is produced by a string 3 times as long as the original fundamental.

Just as a shorter length resulted in a higher pitch, a longer length results in a lower pitch.

This new string does not vibrate 3 times as fast as the source tone, but rather 1/3 as fast. To justify this tone, we have to multiply it by 2, twice, so that 1/3 becomes 2/3 and then 4/3, becoming higher in pitch with each duplication.

Just as the first new pitch 3/2 was initially produced *above* the source tone, this
new pitch 4/3 was initially produced *below* it. This *above* versus *below*
relationship is called *inversion*. Just as 3/2 is shown with an arrow pointing to the
right from the lower tonic, its inversion 4/3 is shown with an arrow pointing in the opposite
direction from the higher tonic.

The figure below shows the symmetrical relationship between 3/2 and 4/3. These two tones,
3/2 and 4/3, are *inversions* of one another.

4/3 is equal to 1 and 1/3 or 1.33 stated as a decimal fraction, shown on the common number line below.

Just as the first new tone 3/2 is derived *above* the tonic 1/1, so the symmetrical
tone 4/3 is derived *below* the tonic 2/1. The name given to this second new tone is the
same as the first, with the prefix *sub* which simply means *below*; it is called
the *subdominant*.

The figure below summarizes the names of the tones we have derived so far.

Since 2/1 is a duplication of 1/1, we have three *non duplicate* tones. These three
tones form the backbone of the Western tonal system. For this reason, the *intervals*,
or distances between the tonic and each of the other tones, are called *Perfect*.
No other intervals receive this epithet.

Before moving on, we can review how the system has been derived so far. We have used a string on a viola, bowing the string at its entire length, at 1/2 of this length, at 1/3 of this length, and at 1/4 of its entire length.

The tones thus produced are the first four harmonics, numbered 1, 2, 3 and 4, of the harmonic
series. Only the third harmonic results in a tone which is not a *duplication* of the first
harmonic, which is the *source tone* or *fundamental*. This relationship is used to
build two new tones which are symmetrically justified around the source tone. These two symmetrical
tones together with the source tone constitute the three most important tones of the Western system.

The importance of these tones is reflected in the names music theorists have given to them.
The names given here were popularized during the 18th century and are standard today. To summarize,
the source tone is called *tonic*, which means *tone* in Greek, the first new tone above
the tonic is called the *dominant*, and its symmetrical counterpart below the tonic is called
the *subdominant*, meaning the *dominant below*. The dominant and subdominant are
*inversions* of each other. The names *tonic*, *dominant* and *subdominant*
logically reflect both the derivation and the function of each tone as the most important in the
Western system.

The figure below shows overlapping patterns for the first four harmonics. This figure can serve as a reminder that when the entire length of a string is made to vibrate without touching the string at any point, the string vibrates in all its harmonics at once.

Although the freely vibrating string produces harmonics well beyond 4, the first four harmonics are all we need to complete our system. Since 3 is the only tone within this limit which is a new pitch, we call this a 3 limit system.

How can we produce more tones without going beyond the 3-limit? Try to answer this question on your own before going on.

NEXT: 5 & 7 Tones